Radiation of an electron in an infinitely long waveguide

Abstract

This paper deals with the electromagnetic fields excited by a single electron with prescribed motion in an infinitely long cylindrical space surrounded by perfectly conducting walls, and filled with a uniform linear medium. The medium is assumed to be lossy in order to avoid complexities arising from the radiation condition. The problem is attacked by first obtaining a general expression for the electric Hertz vector. From this Hertz vector the electric fields are calculated for (1) uniform motion of the electron parallel to the axis of a waveguide of arbitrary cross section (2) uniform but longitudinally oscillatory motion and (3) undulating motion with constant velocity parallel to the axis of a rectangular wave-guide. It is found that the electric field can be represented as the sum of residues corresponding to different field patterns. The power flow from the electron to the electromagnetic field is calculated. Čerenkov and undulator radiation are not the only causes of energy loss. In the presence of a dielectric medium power also flows into “electrostatic oscillations.” Expressions for the energy loss of an electron similar to those previously given by Bohm and Pines and by Fermi are obtained by our method. It is shown that an upper limit exists for the energy loss including Čerenkov and “electrostatic” loss. No such upper limit can be given in the case of undulator radiation. This is shown to be connected with the fact that the latter radiation is propagated both forward and backward. In the limit of infinitely large cross section of the waveguide the formulae tend to the results for the unbounded medium. This shows that the influence of the walls is correctly calculated.